F303 Class 05

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Published on April 28, 2008

Author: Nellwyn

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F303 – Intermediate Investments:  F303 – Intermediate Investments Class 5 Asset Pricing Models: Capital Asset Pricing Model Andrey Ukhov Kelley School of Business Indiana University Outline of This Class:  Outline of This Class Why we need asset pricing models Capital Asset Pricing Model (CAPM) Implications of CAPM for investors Empirical evidence on CAPM Minimum-Variance Frontier:  Minimum-Variance Frontier E(R) Asset Pricing Models:  Asset Pricing Models These are equilibrium models that describe why assets having different characteristics generate different expected returns. Useful to generate expected returns; investors will want to have models that generate benchmark returns. Examples: Capital Asset Pricing Model (CAPM); Arbitrage Pricing Theory (APT). Brief Overview of the CAPM:  Brief Overview of the CAPM The Capital Asset Pricing Model (CAPM) is a centerpiece of finance. This model generates an exact prediction of the risk-return relationship. Why is this important? CAPM serves as a benchmark…for any asset we would need to have a view of the “fair” return given the asset’s risk. CAPM – The Assumptions:  CAPM – The Assumptions Remember that the CAPM is about the equilibrium expected returns of risky assets. We have a hypothetical world and ask the question: “what if”? Assumptions: Mean and variance are the only factors that matter; All assets are divisible; All investors plan for one holding period; No transaction costs and no taxes; Perfect competition; All investors are rational mean-variance optimizers; Risk-free asset available. Equilibrium Returns:  Equilibrium Returns We observe that average returns vary across assets. Why is it that GM generates a different return that Lucent and Delta Airlines? Can we explain this? The CAPM argues that these variations occur because of the assets’ Betas. What does it mean to be in equilibrium? In such a state, the risk premium per unit of risk is the same for all assets. Deriving the CAPM:  Deriving the CAPM If all investors use the same Markowitz analysis and apply it to the same universe of securities, using the same inputs and economic information for the analysis and do such a task for the same time period… …then they must reach the same conclusion in identifying the optimal risky portfolio – which we call the Market Portfolio – which is the tangent portfolio between the Capital Market Line and the Efficient Frontier. Mean-Standard Deviation Frontier:  Mean-Standard Deviation Frontier Capital Allocation Line (CAL) E(R) Efficient Frontier M Market Portfolio Security Market Line:  Security Market Line Security Market Line (SML) E(R) Market Portfolio Beta and Expected Return:  Beta and Expected Return SML E(R) CAPM’s Major Conclusions :  CAPM’s Major Conclusions The Expected Return on an asset can be expressed as: …giving us a linear relationship between the Expected Return and Beta. Not all risk is compensated…an asset’s expected return is ONLY related to its level of systematic risk, given by its Beta. Expected Return on Stocks E(R)=5% + Beta(13.6% - 5%):  Expected Return on Stocks E(R)=5% + Beta(13.6% - 5%) The CAPM: Its Implications:  The CAPM: Its Implications Investors will remove firm-specific risks by diversifying across different industrial sectors; But, even the most diversified portfolio will be risky (Market Risk cannot be diversified away); Investors will be rewarded for investing in such a risky portfolio by earning excessive returns (portfolio returns less risk free rate); The returns from a specific investment (or asset) depend exclusively on the extent to which that investment (or asset) affects the Market Risk…and that is captured by Beta. Market Portfolio:  Market Portfolio What is the Market Portfolio? Summing over all the portfolios of all the investors will give you the aggregate risky portfolio…which is equal to the entire wealth of the economy. This should give you the market portfolio [referred to as M]. What is the presence of individual stocks in M? The proportion of each stock in M is equivalent to the stock’s market value divided by the entire market capitalization. Market Price of Risk:  Market Price of Risk The Market Portfolio has a risk premium and a variance of , giving us the Reward-to-Risk ratio of: This is the market price of risk. It quantifies the excessive return that investors demand to take on the portfolio risk. This number will give you the risk premium that should be earned per unit of portfolio risk. Security Market Line:  Security Market Line The expected return-beta relationship is captured graphically by the Security Market Line (SML). Remember: “fairly priced” assets must fall exactly on the SML! The market beta is equal to 1; hence from the SML (where Beta=1) we can get the expected return from the Market Portfolio. The SML provides a benchmark for the evaluation of investment performances. Mispriced Securities: What Should Happen?:  Mispriced Securities: What Should Happen? SML E(R) Security A Security B How Do We Get Beta?:  How Do We Get Beta? The main insight: the correct risk premium on an asset is determined by its contribution to the risk of the Market Portfolio. This contribution is the asset’s Beta. Starting point: Let us consider one stock, Delta. What is Delta’s contribution to the variance of Market Portfolio? How Do We Get Beta?:  How Do We Get Beta? Now suppose that the investor, who is invested 100% in the Market Portfolio, decides to increase his position in Delta by a very small fraction . He/She finances his/her purchase of by borrowing at the risk-free rate. What is he/she getting in terms of returns? There is the original position in the Market Portfolio, plus a negative position of size in the risk-free asset giving , and a long position of size in Delta that will return . How Do We Get Beta?:  How Do We Get Beta? What is the new portfolio’s excess returns? What is the variance of the new portfolio? And what is the increase in the variance? How Do We Get Beta?:  How Do We Get Beta? Hence, the marginal price of risk of Delta is given by: In equilibrium, the marginal price of risk of Delta must be equal to that of the Market Portfolio. This gives us the following: How Do We Get Beta?:  How Do We Get Beta? To get the fair risk premium of Delta, we have: The ratio is the contribution of Delta to the variance of the Market Portfolio. This is Beta! The expected return-beta relationship of the CAPM: Estimating Betas:  Estimating Betas But in practice, how can we estimate betas? How do we use them for security analysis? One possible answer is a regression analysis. Consider a sample of returns observed for a period of months (or weeks, etc) for t=1,2,3,…T Let us denote the returns on security i, the market and the risk free asset respectively. Estimating Betas:  Estimating Betas One standard method is to estimate Beta through the characteristic line as follows: You would need to use monthly data spanning 5 years…giving you a total of 60 observations. Then you use the excess returns of an individual security as the dependent variable and the excess return from the market as the independent variable as inputs in the regression model. Estimating Betas: An Example:  Estimating Betas: An Example Consider, as one example, the GM data in the book BKM (chapter 8) to estimate Beta. A simple regression in Excel would generate the following results: Where the R-squared is 0.575. What does it mean to have a Beta of 1.1355? What is the implication of such a result to investors? Regression Analysis to Get Beta:  Regression Analysis to Get Beta Return on the Market Return on Individual Security January 1996 February 1996 March 1996 April 1996 Slope = Beta Industry Asset Betas [Obtained from D. Mullins, “Does the CAPM Work?”, Harvard Business Review, vol. 60, pp. 105-114]:  Industry Asset Betas [Obtained from D. Mullins, “Does the CAPM Work?”, Harvard Business Review, vol. 60, pp. 105-114] Estimating Betas: Commercial Supplies:  Estimating Betas: Commercial Supplies Value Line employs 5 years of weekly data and value-weighted NYSE as the market. Bloomberg employs 5 years of monthly data and S&P 500 as the market. BARRA Securities’ Alphas:  Securities’ Alphas The security’s Alpha, , is the difference between the expected returns predicted by the CAPM and the actual returns. Nonzero alphas mean that securities do not plot on the SML. Example: Let us say that the expected market return is 14%, risk free rate is 6% and that a stock has a beta of 1.2. Then the SML would predict the stock’s return to be: 6% + 1.2(14% - 6%) = 15.6% If during the holding period, the stock produced a actual return of 18%, then the security’s alpha is 2.4% Identifying Mispriced Assets:  Identifying Mispriced Assets One possible use of the CAPM is security analysis: uncovering securities with nonzero alphas. If , then the asset’s expected return is too high (low) according to the CAPM and is under priced (overpriced). This is referred to as an abnormal or risk-adjusted return. The problem: different than zero could be either produced by a mis-specified CAPM or an inefficient market (this is the so-called joint hypothesis problem) Example of Mispriced Assets:  Example of Mispriced Assets Have a look at these average annualized returns for the last 15 years for the following three portfolios: Franklin Income Fund 12.9% Dow Jones Industrial Average 11.1% Salomon’s High Grade Bond Index 9.2% Now assume that the expected return on the Market Portfolio is 13% and that the risk free rate is 7%. In addition, suppose that the funds’ Betas are as follows: Franklin Income Fund 1.0 Dow Jones Industrial Average 0.683 Salomon’s High Grade Bond Index 0.367 Example of Mispriced Assets:  Example of Mispriced Assets Now, let us calculate, using the information given before, the Expected Return using the CAPM: Franklin Income Fund 13.0%=7%+1.0 x (13%-7%) Dow Jones Industrial Average 11.1%=7%+0.683 x (13%-7%) Salomon’s High Grade Bond Index 9.2%=7%+0.367 x (13%-7%) Interpretation: For the Dow Jones and the Salomon’s Bond, the returns implies from the CAPM are in line with those observed in the last 15 years. For the Franklin Fund, the market was expecting 13% and got less than that: this means that there was an abnormal return. On a risk-adjusted basis, the fund has underperformed. How Good is the CAPM in Predicting Returns?:  How Good is the CAPM in Predicting Returns? Let us have a look at the literature on anomalies in the stock market…that is patterns of returns that cannot be explained by the CAPM; Then we introduce the CAPM debate, which was started by Roll (1977). Here, there are theoretical issues and empirical issues. Stock Anomalies: Small Firm Effect:  Stock Anomalies: Small Firm Effect Small market capitalization firms have produced higher average returns than was predicted by the CAPM. Banz (1981) and Reinganum (1981) use monthly data and daily data respectively and find this result. The effect is strongest for the month of January. What could explain this? Liquidity? Small firms are less liquid (the ability to buy or sell at reasonable prices and time) than large firms and this could be driving these higher returns. Stock Anomalies: January Effect:  Stock Anomalies: January Effect January has historically produced higher returns than other months during the year. The effect is particularly strong for small firms. Tax-loss selling could be one possible explanation. But the effect persists in international markets where capital gains tax does not exist. “Window dressing” by fund managers and institutional investors. There is also the “Day-of-the-Week” effect: stock returns are lower over the weekend (returns are negative on Mondays, but they are positive from Wednesday to Friday). Average Daily Returns (1928 – 1982 on NYSE):  Average Daily Returns (1928 – 1982 on NYSE) Price/Earnings Ratio:  Price/Earnings Ratio Evidence that securities with low Price/Earnings (P/E) ratio have higher average returns. Basu (1977) explained violations from the CAPM by using P/E ratios…for a sample of NYSE securities there was a clear negative relationship between P/E ratios and the average returns in excess of those predicted by the CAPM. Following a very simple strategy of buying the quintile of smallest P/E securities and selling short the top quintile, would have produced an average abnormal return of 6.75% (annual, from 1957 to 1971). The CAPM Debate:  The CAPM Debate Remember what we are trying to verify: The CAPM gives us a linear relationship between an asset’s expected return and its Beta. The question: Is this what we get in real life? We first review Roll’s critique and then proceed to review empirical evidence in favor of CAPM (early 1970s) and against it (starting in the 1980s and getting big in 1990s). Roll’s Critique (1977):  Roll’s Critique (1977) Roll states that the only acceptable test of the CAPM is whether the market portfolio is mean-variance efficient. But, the market portfolio is, technically speaking, a portfolio that includes ALL the assets in the economy (listed and unlisted stocks, listed and unlisted bonds, property, human capital, etc.). Roll’s Critique (1977):  Roll’s Critique (1977) In empirical tests, we do not use the true market portfolio because there is no data…we have to settle for a proxy, like the Dow Jones Industrial Index, the FTSE All Share Index, etc. If performance is measured relative to a proxy that is ex post efficient, then no security will produce abnormal performance… On the other hand, if performance is measured relative to an ex post inefficient proxy, then any ranking could be possible… Roll’s Critique (1977):  Roll’s Critique (1977) What does this mean? Is this just a quibble? No! Actually, it is very important… A small change in the proxy of the market portfolio – for example, going from S&P 500 to the Wilshire list of 5,000 listed securities – can alter dramatically the expected returns! Since no one knows the true market portfolio then nobody can conclude whether the CAPM holds or not. First Empirical Evidence:  First Empirical Evidence Three researchers, Black, Jensen and Scholes, way back in 1972, wanted to test the CAPM’s predictions. They divided the NYSE stocks into ten portfolios. The first portfolio was formed from securities with the lowest Betas, the second contained the next 10% with the next lower Betas, so on and so forth, up to the tenth portfolio. The study was carried out over a 35 year period. Black, Jensen and Scholes (1972):  Black, Jensen and Scholes (1972) The evidence shows that there was an exact straight-line relationship between a portfolio’s Beta and the average return. This would be in line with the CAPM’s predictions. The same was found in empirical studies by Fama and MacBeth (1973) and Blume and Friend (1973). What Happened Next…?:  What Happened Next…? Toward the end of the 1970s we had some less favorable evidence coming out against the CAPM. This gave rise to the anomalies literature, mentioned before. Basu (1977) reported the Price/Earnings effect. Then Banz (1981) found the small size effect where firms with low market capitalization generate higher returns than predicted by the CAPM. The Death of Beta:  The Death of Beta Fama and French (1992 and 1993) show that Beta is flat…has no power. This has led people to declare that “Beta is Dead!” Fama and French show that Beta cannot explain the difference in returns formed on the basis of the Book-to-Market ratio. Firms with high book-to-market ratios generate higher returns than predicted by the CAPM. The Death of Beta:  The Death of Beta In conclusion, Fama and French (1992) find that just two variables – market equity (the firm’s size) and the ratio of the book equity (the book value of the equity) to market equity (equity’s value on the market) – capture much of the cross-section of average stock returns. There seems to be no role for Beta to explain returns! Something to Remember…:  Something to Remember… Let us say that you hear a fund manager saying: "I have followed the CAPM and purchased high Beta securities last year; but they did worse than low Beta securities last year! I say that, based on this evidence, the CAPM is dead.” Would you agree with the fund manager? Is this a valid test of the CAPM’s validity? Defending the CAPM…:  Defending the CAPM… Problems with data snooping and sample selection biases; If Beta died…it died very recently (earlier work shows that the CAPM holds); The results from the anomalies literature could indicate significant deviations from the CAPM, but there is little theoretical motivations for these results…we do not have any model for the behavior found in the anomalies literature. Key Points to Remember:  Key Points to Remember Major assumptions and conclusions from the CAPM…remember that the model gives us a linear relationship between the Expected Return and Beta. Not all risk is compensated…an asset’s expected return is ONLY related to its level of systematic risk, given by its Beta. Meaning of Beta; Significance of the Market Portfolio. Key Points to Remember:  Key Points to Remember The Security Market Line (SML) and how mispriced securities behave…why in equilibrium assets should plot on the SML. Security’s Alpha and its meaning. Empirical evidence against and in favor of the CAPM. The main result is that some studies have recently found that Beta could have no explanatory power…but this has been disputed by other studies.

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